# Convergence Rate Bounds for the Mirror Descent Method: IQCs, Popov Criterion and Bregman Divergence

arXiv: https://arxiv.org/abs/2304.03886 · HTML: https://mlsanity.com/paper/2304.03886/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Apr 2023.

## Built on by (most cited first, 1 of 1)

- [A Closed-Form Upper Bound for Admissible Learning-Rate Steps in Belief-Space Dynamics](https://mlsanity.com/paper/2605.06741/) (2026)

## Abstract (arXiv)

This paper presents a comprehensive convergence analysis for the mirror descent (MD) method, a widely used algorithm in convex optimization. The key feature of this algorithm is that it provides a generalization of classical gradient-based methods via the use of generalized distance-like functions, which are formulated using the Bregman divergence. Establishing convergence rate bounds for this algorithm is in general a non-trivial problem due to the lack of monotonicity properties in the composite nonlinearities involved. In this paper, we show that the Bregman divergence from the optimal solution, which is commonly used as a Lyapunov function for this algorithm, is a special case of Lyapunov functions that follow when the Popov criterion is applied to an appropriate reformulation of the MD dynamics. This is then used as a basis to construct an integral quadratic constraint (IQC) framework through which convergence rate bounds with reduced conservatism can be deduced. We also illustrate via examples that the convergence rate bounds derived can be tight.